mc_sgd
======

.. py:function:: t3toolbox.backend.optimizers.mc_sgd(problem, x0, rng, batch, draw = None, max_iter = 3000, check_every = 25, smooth_tau = 2.0, plateau_lag = 4, use_jit = False)

   .. code-block:: python

      def mc_sgd(
              problem:     Problem,    # the fixed-rank least-squares problem
              x0:          Tangent,    # initial cores (U, G)
              rng,                     # np.random.Generator -- passed to the draw each step
              batch:       int,        # measurements per minibatch (for the default flat draw; ignored if draw given)
              draw:        typ.Optional[typ.Callable] = None,  # custom draw(rng)->(sample_B,data_B); None = flat default
              max_iter:    int   = 3000,
              check_every: int   = 25,   # iterations between full-batch loss checks (the absolute-iteration window)
              smooth_tau:  float = 2.0,  # EMA timescale of the loss check, in checks
              plateau_lag: int   = 4,    # stop when the smoothed loss rises over this many checks
              use_jit:     bool  = False,  # jit the per-step kernel (gradient + Cauchy step + retract) when jax
      ) -> typ.Tuple[Tangent, dict]:     # (x_cores, stats)


   Manifold Cauchy SGD (T4S §5.3.2): minibatch + the tuning-free Cauchy step
   ``α = ‖g‖² / ‖𝒥g‖²`` -- no learning rate. Intended for the **manifold** geometry (its bounded
   retraction makes the raw Cauchy step stable; on the additive corewise chart use ``adam``). Stops on an
   exponentially-smoothed **full-batch** loss with an **absolute-iteration window** (``plateau_lag *
   check_every`` iterations) -- decoupled from batch size (the epoch-based window made small batches
   fragile; findings in a separate research repo, maintainer-local). ``use_jit`` jits the per-step kernel (the host loop
   draws minibatches; the full-batch stop check stays on the host), auto-converting numpy inputs to jax
   first (:py:func:`_prepare_jit_inputs`) -- so the **default** ``flat_draw`` then slices the on-device
   ``sample`` / ``data`` (a custom ``draw`` should return jax minibatches to stay on device).